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Abstract

In this article, we deal with new properties of a Solid Core Photonic Bandgap (SC-PBGF) fiber with intersticial air holes (IAHs) in its transverse structure. It has been shown recently, that IAH enlarges its bandgaps (BG), compared to what is observed in a regular SC-PBGF. We shall describe the mechanisms that account for this BG opening, which has not been explained in detail yet. It is then interesting to discuss the role of air holes in the modification of the Bloch modes, at the boundaries of the BG. In particular, we will use a simple method to compute the exact BG diagrams in a faster way, than what is done usually, drawing some parallels between structured fibers and physics of photonic crystals. The very peculiar influence of IAHs on the upper/lower boundaries of the bandgaps will be explained thanks to the difference between mode profiles excited on both boundaries, and linked to the symmetry / asymmetry of the modes. We will observe a modification of the highest index band (nFSM) due to IAHs, that will enable us to propose a fiber design to guide by Total Internal Reflection (TIR) effect, as well as by a more common BG confinement. The transmission zone is deeply enlarged, compared to regular photonic bandgap fibers, and consists in the juxtaposition of (almost non overlapping) BG guiding zones and TIR zone.

Figures (7)

Panel (a) shows the cladding we study. In an elementary cell (inside the white bordered parallelogram), two IAHs (in blue) surround a parabolic radial index profile, that model a high index rod (in dark red), with d/Λ = 0.725. The maximum index gradient, between the rod center and the background is Δn = 3.2∙10-2. IAHs have a diameter dair/Λ = 0.152. Panel (b) shows the BG diagram for the cladding with IAHs (full thick black line and dashed thick black line, which represents the FSM line), and without IAH (grey line with symbols). The thin horizontal black line represents the silica index nc = 1.45.

Band diagram for structure B cladding. Only the lines corresponding to Γ, M or K have been represented. The BG have been shaded, and isolated rod LP modes are labeled. An insert represents the Brillouin zone of an hexagonal lattice. Particular values of β have been pointed out, in units of Λ-1.

Intensity and field profile for both borders of the first beam of permitted modes, associated to the LP01 mode, computed near the tip of first BG, at βΛ = 3, cf. Fig. 2, for structure B. Color bar at the right of the panels refer to the normalized field or intensity value. A contour plot on Fig. 3(b) shows iso-intensity lines, in logarithmic scale. The corresponding color bar is on the left of panel (b). On every panel, the dotted circles represent the high index inclusions. The arrow on Fig. 3(c) indicate ΓK direction.

Real part of one transverse component of the electric field, for both borders of the second beam of permitted mode, associated to the LP11 mode, computed at βΛ = 7, cf. Fig. 2, for structure B. The arrow on Fig. 4(b) indicates the ΓM direction.

Real part of one transverse component of electric field, Ex, for both borders of the second beam of permitted mode, associated to the LP11 mode, computed at βΛ = 7, cf. Fig. 2, for structure A and B. The dotted circles represent the geometry of the structure.

Panel (a) shows a zone around the core of the fibre we study. The color code refers to the different index of the constituent – Ge-doped silica, silica, air. The boundaries are those of the periodic cladding without defect. On panel(b), one can see (lines without symbols) the dispersion diagram of the lowest frequency band index – nFSM – of the cladding structure. Different sizes of IAH have been used. From top to bottom, the dashed lines correspond to IAH radius of rIAH = 2.5.10-2 and rIAH = 4.5.10-2, in Λ units. The full black line corresponds to structure A, rIAH = 7.5.10-2, as used in section (2). The line with symbols corresponds to the TIR mode observed in the fiber we study.

Tables (2)

Table 2. Synthetic presentation of the properties of permitted mode borders. The nature of the border, as well as the influence of the presence (or absence) of IAH have been mentioned, due to constructive (or destructive) interference between the rods.

Metrics

Table 1.

Summary of the characteristics of the first four BG boundaries.

delimiting modes

upper mode border

lower mode border

BG

rod mode

band type

rod mode

band type

BG I

LP01

K

LP11

M

BG II

LP11

Γ

LP02

Γ

BG III

LP21

M

LP12

M

BG IV

LP31

K

LP03

Γ

Table 2.

Synthetic presentation of the properties of permitted mode borders. The nature of the border, as well as the influence of the presence (or absence) of IAH have been mentioned, due to constructive (or destructive) interference between the rods.

mode position

upper mode border

lower mode border

mode parity

odd mode

M or K

Γ

constructive

destructive

AFFECTED

unaffected

even mode

Γ

M or K

constructive

destructive

AFFECTED

unaffected

Tables (2)

Table 1.

Summary of the characteristics of the first four BG boundaries.

delimiting modes

upper mode border

lower mode border

BG

rod mode

band type

rod mode

band type

BG I

LP01

K

LP11

M

BG II

LP11

Γ

LP02

Γ

BG III

LP21

M

LP12

M

BG IV

LP31

K

LP03

Γ

Table 2.

Synthetic presentation of the properties of permitted mode borders. The nature of the border, as well as the influence of the presence (or absence) of IAH have been mentioned, due to constructive (or destructive) interference between the rods.